Abstracl-A nonlinear mathematical model of a feed-batch fermentation process of Bacillus lhuringiensis (BI.), is derived and a theorem proof of the existence of positive solution of the obtained model, is done. The obtained model is validated by experimental data. An identification and adaptive neural control scheme of the system, represented by a neural identifier and a neural controller, based on the recurrent trainable neural network model, is proposed. The applicability of the proposed adaptive control scheme is confirmed by simulation results which exhibits a good convergence.Index Terms-Adaptive control, fed-batch fermentation of Bacillus lhuringiensis, dynamic backpropagation, recurrent neural networb.

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Biological treatment is attractive as a potentially low-cost technology, which converts toxic organic contaminants into CO 2 and biomass. Since the 70's, this technology has been applied for the hydrocarbon degradation, and today, it is considered as the best alternative to clean up polluted soils. For this bioprocess, one challenge is to provide enough O 2 and nutrients to enable rapid conversion of contaminants by either indigenous microorganisms or inoculated species. Another challenge is to achieve efficient contact between the active microorganisms and the contaminants, which may represented a problem with in-situ treatment. An attractive alternative to overcome this problem is to apply a biological treatment in slurry phase using Horizontal Rotating Drum (HRD). Nowadays, semi empirical HRD models, based on the Monod equation, have been developed to predict microorganism growth as a function of available contaminants concentration. However, as the application of such models requires experimental work for calculating the kinetics parameters involved, so an alternative modeling technique is required. The Kalman Filter Recurrent Neural Network Model (KF RNNM) offers many advantages as the possibility to approximate complex non linear high order multivariable processes, as the biodegradation process is. The KF RNNM has been applied for measurement data filtering and parameters plus state estimation of hydrocarbons biodegradation process, contained in polluted slurry, treated in a rotating bioreactor. Then the KF RNNM is simplified and used to design a Sliding Mode Control (SMC) of two-input two-output high order nonlinear plant. The KF RNNM learning algorithm is the dynamic Backpropagation one (BP). The graphical simulation results of the system approximation, and indirect adaptive neural control, exhibited a good convergence, and precise reference tracking.

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